By Alexander S. Holevo

This ebook is dedicated to elements of the rules of quantum mechanics during which probabilistic and statistical techniques play a vital function. the most a part of the e-book issues the quantitative statistical concept of quantum dimension, in accordance with the idea of confident operator-valued measures. prior to now years there was great development during this course, encouraged to a good volume by way of new functions resembling Quantum Optics, Quantum conversation and high-precision experiments. The questions of statistical interpretation, quantum symmetries, idea of canonical commutation kin and Gaussian states, uncertainty kin in addition to new basic bounds about the accuracy of quantum measurements, are mentioned during this ebook in an obtainable but rigorous approach. in comparison to the 1st version, there's a new complement dedicated to the hidden variable factor. reviews and the bibliography have additionally been prolonged and up-to-date

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It follows that B A = 0 and B A = 0. 28) 0 ≤ Mv ≤ I − Mu when u = v. 4 with A = Mu , B = Mv , C = I − Mu . Thus, the formal quantum analog of classical deterministic measurements is the orthogonal resolutions of identity {E u }: E u E v = δuv E u , E u = I. u 26 Alexander Holevo The corresponding quantum measurements are called simple. Whereas a classical deterministic measurement deÀnes a decomposition of the phase space into mutually disjoint sets (u) , a simple quantum measurement furnishes a decomposition of the unitary vector space H into the direct sum of mutually orthogonal subspaces H(u) = E u (H ).

A reader can check that for the Ànite-rank operators ∗ |ϕ j )(ψ j | = j |ψ j )(ϕ j |. , (U ϕ|U ψ) = (ϕ|ψ); ϕ, ψ ∈ H. 5) the last relation can be written as U ∗ U = I, where I is the unit operator, mapping |ψ) into |ψ), ψ ∈ H. An isometric operator mapping H onto H is called unitary. The condition of unitarity is U ∗U = UU ∗ = I. , X ∗ = X . There is a polarization identity which linearly expresses a Hermitean form through its “diagonal” values for ϕ = ψ; therefore a Hermitean operator X is uniquely deÀned by the values (ψ|X ψ), ψ ∈ H.

19). Thus S(H ) is a convex set.

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